Iterative Identification and Restoration of Images

1 The Image Identification and Restoration Problem.- 1.1 Introduction.- 1.2 Restoration Methods.- 1.3 Identification Methods.- 1.4 Scope of the Monograph.- 2 Image Formation Models.- 2.1 Blur Models.- 2.1.1 Linear Image Formation.- 2.1.2 State-Space Representation.- 2.1.3 Boundary Value Problem.- 2.2 Image Models.- 2.2.1 2-D AR Modeling.- 2.2.2 State-Space Representation.- 2.2.3 Model Fitting Problem.- 2.3 Common Point-spread Functions.- 2.3.1 Linear Motion Blur.- 2.3.2 Uniform Out-of-Focus Blur.- 2.3.3 Atmospheric Turbulence Blur.- 2.3.4 Scatter Blur.- 3 Regularized Image Restoration.- 3.1 Ill-Conditionedness of the Image Restoration Problem.- 3.2 Stochastic Restoration.- 3.2.1 Bayesian Methods.- 3.2.2 Wiener Filtering.- 3.2.3 Kalman Filtering.- 3.3 Algebraic Restoration.- 3.3.1 Tikhonov-Miller Regularization.- 3.3.2 Choice of the Regularizing Operator.- 3.3.3 Formal Relation with Stochastic Restoration.- 3.4Multiple Constraints Restoration.- 3.4.1 Deterministic A Priori Constraints.- 3.4.2 Projections onto Convex Sets.- 4 Iterative Image Restoration.- 4.1 VanCittert's Iteration.- 4.1.1 Formulation of the Algorithm.- 4.1.2 Convergence Analysis.- 4.1.3 Reblurring Procedure.- 4.2 Regularization via Truncated Iterations.- 4.3 Iterative Tikhonov-Miller Solution.- 4.4 Implementations with Faster Convergence.- 4.4.1 Analysis of Convergence Speed.- 4.4.2 Method of Conjugate Gradients.- 4.4.3 Iteration Method with Higher Convergence Order.- 5 Image Restoration with Ringing Reduction.- 5.1 Analysis of Ringing Artifacts.- 5.1.1 The Error Spectrum.- 5.1.2 Relation between the Error Spectrum and Ringing Artifacts.- 5.1.3 Ringing Reduction Methods.- 5.2 Constrained Adaptive Iterative Restoration.- 5.2.1 Introduction.- 5.2.2 A Priori Knowledge.- 5.2.3 Formulation of the Algorithm.- 5.3 Conjugate Gradients-based Implementation.- 5.4 Experimental Restoration Results.- 6 Maximum Likelihood Image Identification.- 6.1 Conventional Identification Methods.- 6.2 Maximum Likelihood Estimator.- 6.2.1 Introduction.- 6.2.2 Definition of the Likelihood Function.- 6.2.3 Properties of the Estimator.- 6.2.4 Analytic Solutions.- 6.3Implementations for Noiseless Data.- 6.3.1 Least-Squares Solution.- 6.3.2 Parallel 1-D Least-Squares Solution.- 6.4Implementations for Noisy Data.- 6.4.1 Gradient-based Iterative Optimization.- 6.4.2 Prediction Error Based Solution.- 7 Image Identification Using the EM-Algorithm.- 7.1 Review of the EM-Algorithm.- 7.2 EM-Algorithm Applied to Image Identification.- 7.3 The E-step of the Algorithm.- 7.4 The M-step of the Algorithm.- 7.4.1 Image Model Identification.- 7.4.2 Blur Model Identification.- 7.5 Experimental Results.- 7.5.1 Linear Motion Blur.- 7.5.2 Defocusing Blur.- 8 Methods for Improved Image Identification.- 8.1 Parametric Image Identification.- 8.1.1 Parametric Modeling.- 8.1.2 Image Model.- 8.1.3 Blur Model.- 8.2 Experimental Results Using Parametric Models.- 8.2.1 Linear Motion Blur.- 8.2.2 Atmospheric Turbulence Blur.- 8.2.3 Photographic Motion Blur.- 8.3 Hierarchical Image Identification.- 8.3.1 Use of Resolution Pyramids.- 8.3.2 Downsampling of Blurred Images.- 8.3.3 Image and Parameter Interpolation.- 8.3.4 Decision Tree for PSF Support Size.- 8.4 Experimental Results Using the Hierarchical Method.- 8.4.1 Linear Motion Blur.- 8.4.2 Defocusing Blur.- 8.4.3 Photographic Out-of-Focus Blur.- A Eigenvalue Analysis for 2-D Systems.- B Properties of the Iteration (5.21).- C Derivation of Equation (7.14).